%\onlyShort{\vspace{-0.20in}}
\section{Concluding Remarks and Open Problems}
\label{sec:conc}
\onlyShort{\vspace{-0.1in}}
 
Our work shows that while some local symmetry breaking problems such as coloring and maximal matching can be solved in polylogarithmic rounds in both the LOCAL and CONGEST models, for many others such as MIS, hitting set, and maximal clique it remains a challenge to obtain polylogarithmic time algorithms in the CONGEST model. This dichotomy manifests in hypergraphs
of higher dimension. Understanding this dichotomy can be helpful to make further progress 
in improving the bounds or showing lower bounds, especially in the CONGEST model. In particular, an important open question is whether we can show super-polylogarithmic lower bounds
for MIS for hypergraphs of high dimension in the CONGEST model?

Our results also have implications to solving hypergraph problems in the classical PRAM model.
Our CONGEST model algorithms can be translated into PRAM algorithms running in (essentially) the same number of rounds (up to polylogarithmic factors).
 In particular,  improving over the $\tilde O(\Delta^{o(1)})$ round algorithm for MIS in the CONGEST model can point to better PRAM algorithms for MIS which has been eluding researchers till now. A major question is whether $O(\polylog n)$ or even $O(\polylog m)$  round algorithms are possible
 in the CONGEST model for MIS (as shown here, the answer is ``yes'' in the LOCAL model).

Another aspect of this work, which was one of our main motivations, is using hypergraph algorithms for  solving problems in graphs efficiently.  In particular, our hypergraph MIS algorithm leads to  fast distributed algorithms for the  BMDS and the MCDS problems.
%Improved algorithms for hypergraph problems will lead to improved algorithms for the BMDS and the MCDS problems.
In particular, it will be interesting to see if one can give an algorithm for MCDS that essentially matches
the lower bound of $\tilde{\Omega}(D + \sqrt{n})$ (when $D$ is large).
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